13.12.11 problem 11

Internal problem ID [2423]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8, Series solutions. Page 195
Problem number : 11
Date solved : Monday, January 27, 2025 at 05:52:31 AM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} \left (-t^{2}+1\right ) y^{\prime \prime }-t y^{\prime }+\alpha ^{2} y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 65

Order:=6; 
dsolve((1-t^2)*diff(y(t),t$2)-t*diff(y(t),t)+alpha^2*y(t)=0,y(t),type='series',t=0);
 

\[ y = \left (1-\frac {\alpha ^{2} t^{2}}{2}+\frac {\alpha ^{2} \left (\alpha ^{2}-4\right ) t^{4}}{24}\right ) y \left (0\right )+\left (t -\frac {\left (\alpha ^{2}-1\right ) t^{3}}{6}+\frac {\left (\alpha ^{4}-10 \alpha ^{2}+9\right ) t^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 88

AsymptoticDSolveValue[(1-t^2)*D[y[t],{t,2}]-t*D[y[t],t]+\[Alpha]^2*y[t]==0,y[t],{t,0,"6"-1}]
 

\[ y(t)\to c_2 \left (\frac {\alpha ^4 t^5}{120}-\frac {\alpha ^2 t^5}{12}+\frac {3 t^5}{40}-\frac {\alpha ^2 t^3}{6}+\frac {t^3}{6}+t\right )+c_1 \left (\frac {\alpha ^4 t^4}{24}-\frac {\alpha ^2 t^4}{6}-\frac {\alpha ^2 t^2}{2}+1\right ) \]